LINEAR AND NONLINEAR FREE VIBRATION ANALYSIS OF RECTANGULAR PLATE - Adah, Edward; Onwuka, David; Ibearugbulem, Owus; Okere, Chinenye (Journal of Civil Engineering, Science and Technology, 2021)
The major assumption of the analysis of plates with large deflection is that the middle surface displacements are not zeros. The determination of the middle surface displacements, u0 and v0 along x- and y- axes respectively is the major challenge encountered in large deflection analysis of plate. Getting a closed-form solution to the long standing von Karman large deflection equations derived in 1910 have proven difficult over the years. The present work is aimed at deriving a new general linear and nonlinear free vibration equation for the analysis of thin rectangular plates. An elastic analysis approach is used. The new nonlinear strain displacement equations were substituted into the total potential energy functional equation of free vibration. This equation is minimized to obtain a new general equation for analyzing linear and nonlinear resonating frequencies of rectangular plates. This approach eliminates the use of Airy’s stress functions and the difficulties of solving von Karman's large deflection equations. A case study of a plate simply supported all-round (SSSS) is used to demonstrate the applicability of this equation. Both trigonometric and polynomial displacement shape functions were used to obtained specific equations for the SSSS plate. The numerical results for the coefficient of linear and nonlinear resonating frequencies obtained for these boundary conditions were 19.739 and 19.748 for trigonometric and polynomial displacement functions respectively. These values indicated a maximum percentage difference of 0.051% with those in the literature. It is observed that the resonating frequency increases as the ratio of out–of–plane displacement to the thickness of plate (w/t) increases. The conclusion is that this new approach is simple and the derived equation is adequate for predicting the linear and nonlinear resonating frequencies of a thin rectangular plate for various boundary conditions.